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In order to learn how to choose a, we now consider the case of a large enough voltage across the LC cell to fully orient the LC molecules apart from two thin layers on top of the orientation layer, due to Δε > 0 in parallel to the electric field. The linearly polarized light coming in at angle a no longer experiences birefringence, as it is only exposed to the refractive index n┴. It reaches the plane z = d with the phase shift 2π(n┴d/λ). Its component Ep passing an analyser with the angle (π/2) − α in Figure 3.8 is
(3.74)
whereas the component Es passing an analyser with the angle π − α in Figure 3.8 is
(3.75)
The intensities belonging to Ep and Es are
(3.76)
and
(3.77)
The bar over the cos terms means the average over time needed for calculating the intensity. The maximum of Ip occurs for α = π/4, which (according to Figure 3.8) places the polarizer and analyser in parallel. Is assumes a maximum for α = 0, for which again the polarizer and analyser pertaining to Es are in parallel.
In Figures 3.9 and 3.10, the intensities Iy′ and Ix′ in Equations (3.73) and (3.72) are plotted versus x= (πdΔn/λ) and λ = (πdΔn/x). Iy′ in Figure 3.9 becomes zero at
(3.78)
Figure 3.9 The intensity Iy, of the Fréedericksz cell for two values of α in Equation (3.73)
Figure 3.10 The intensity Ix, of the Fréedericksz cell for two values of α in Equation (3.72)
or
from which
follows. The function cos2x is lowered around x = π/2 by the multiplication with sin22α. This is most welcome, as it enhances the black state, which is imperfect by the suppression of only λ0. This is demonstrated by two values for a in Figure 3.9. The intensity Ix′ in Equation (3.72) exhibits the same dependence on x as Iy′, and is plotted in Figure 3.10, demonstrating that at x = π/2
(3.81)
is the maximum intensity independent of α, which passes an analyser placed in the direction x′.
We are now ready to determine the contrast C(α) for the normally black and normally white cell as a function of α. C is defined as
where Lmax is the maximum luminance assumed to be proportional to the maximum intensity, whereas Lmin stands for the minimum luminance assumed to be proportional to the minimum intensity.
We first investigate the normally white mode. In the field-free state, the incoming linearly polarized light with angle α in Figure 3.11(a) again generates linear polarization for a wavelength λ0 at the angle β = π − α in Equation (3.62) with the intensity Ix′ given in Equation (3.72) representing the white state. If a large enough field is applied, the light reaches the analyser linearly polarized in the direction α independent of λ. Hence, the analyser in Figure 3.11(a) allows the component
Figure 3.11 The angles of the electric field and the polarizers in a normally white Fréedericksz cell with linearly polarized light at the output d = λ/2Δn. (a) Crossed polarizers; (b) parallel polarizers
to pass. This represents the black state. From Equations (3.72) and (3.83),
